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Let $\Sigma_g \to E \to \Sigma_h$ be a surface bundle over a surface withmonodromy representation $\rho: \pi_1 \Sigma_h \to\operatorname{Mod}(\Sigma_g)$ contained in the Torelli group $\mathcal{I}_g$.In this paper we express the cup product structure in $H^*(E, \mathbb{Z})$ interms of the Johnson homomorphism $\tau: \mathcal{I}_g \to \wedge^3 (H_1(\Sigma_g, \mathbb{Z}))$. This is applied to the question of obtaining an upperbound on the maximal $n$ such that $p_1: E \to \Sigma_{h_1}, ..., p_n: E \to\Sigma_{h_n}$ are fibering maps realizing $E$ as the total space of a surfacebundle over a surface in $n$ distinct ways. We prove that any nontrivialsurface bundle over a surface with monodromy contained in the Johnson kernel$\mathcal{K}_g$ fibers in a unique way.

Cup products, the Johnson homomorphism, and surface bundles over surfaces with multiple fiberings